1 Effect of UPSTM-Based Decorrelation on Feature Discovery

1.0.1 Loading the libraries

library("FRESA.CAD")
library(readxl)
library(igraph)
library(umap)
library(tsne)
library(entropy)
library(TH.data)
library(psych)
library(whitening)
library("vioplot")
library("rpart")
library(mlbench)

op <- par(no.readonly = TRUE)
pander::panderOptions('digits', 3)
pander::panderOptions('table.split.table', 400)
pander::panderOptions('keep.trailing.zeros',TRUE)

1.1 Material and Methods

Source W. Nick Street, Olvi L. Mangasarian and William H. Wolberg (1995). An inductive learning approach to prognostic prediction. In A. Prieditis and S. Russell, editors, Proceedings of the Twelfth International Conference on Machine Learning, pages 522–530, San Francisco, Morgan Kaufmann.

Peter Buehlmann and Torsten Hothorn (2007), Boosting algorithms: regularization, prediction and model fitting. Statistical Science, 22(4), 477–505.

1.2 The Data

wpbc {TH.data}


data("wpbc", package = "TH.data")
sum(1*(wpbc$status=="R" &  wpbc$time <= 24))
#> [1] 29
wpbc <- subset(wpbc,time > 36 | (status=="R" & time <= 36))
summary(wpbc$time)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>    1.00   36.75   60.50   58.79   78.75  125.00
wpbc$status <- 1*(wpbc$status=="R"  & wpbc$time <= 36)
wpbc <- wpbc[complete.cases(wpbc),]
pander::pander(table(wpbc$status))

0	1
103	34

wpbc$time <- NULL

1.3 Comparing IDeA vs PCA vs EFA

1.3.1 Getting the top correlated variables

#plot(wpbc[,colnames(wpbc)!="status"],col=wpbc$status,cex=0.65,cex.lab=0.5,cex.axis=0.75,cex.sub=0.5,cex.main=0.75)
rawCor <- cor(wpbc[,colnames(wpbc)!="status"])

  gplots::heatmap.2(abs(rawCor),
                    trace = "none",
  #                  scale = "row",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "Original Correlation",
                    cexRow = 0.5,
                    cexCol = 0.5,
                     srtCol=45,
                     srtRow= -45,
                    key.title=NA,
                    key.xlab="Pearson Correlation",
                    xlab="Feature", ylab="Feature")




diag(rawCor) <- 0;
getMaxcor <- apply(rawCor,2,max)
topCorrelated <- getMaxcor[which.max(getMaxcor)]
whotoMax <- getMaxcor[getMaxcor == topCorrelated]
plot(wpbc[,names(whotoMax)],main="Top Correlated variables")

1.3.2 IDeA

IDeAwpbc <- IDeA(wpbc)
#plot(IDeAwpbc[,colnames(IDeAwpbc)!="status"],col=wpbc$status,cex=0.65,cex.lab=0.5,cex.axis=0.75,cex.sub=0.5,cex.main=0.75)
#pander::pander(attr(IDeAwpbc,"UPSTM"))
pander::pander(getLatentCoefficients(IDeAwpbc))

La_mean_perimeter:

mean_radius mean_perimeter mean_compactness

-6.66 1 -33.7
La_mean_area:

mean_radius mean_area

-111 1
La_mean_concavity:

mean_compactness mean_concavity

-1.06 1
La_mean_concavepoints:

mean_concavity mean_concavepoints

-0.445 1
La_mean_fractaldim:

mean_fractaldim worst_fractaldim

1 -0.292
La_SE_perimeter:

SE_radius SE_perimeter

-6.88 1
La_SE_area:

SE_radius SE_area

-151 1
La_SE_concavity:

SE_compactness SE_concavity

-0.886 1
La_SE_fractaldim:

SE_compactness SE_fractaldim

-0.0931 1
La_worst_radius:

mean_radius worst_radius

-1.3 1
La_worst_texture:

mean_texture worst_texture

-1.27 1
La_worst_perimeter:

worst_radius worst_perimeter

-6.63 1
La_worst_area:

worst_radius worst_area

-139 1
La_worst_compactness:

worst_compactness worst_fractaldim

1 -6.34

mean_radius	mean_perimeter	mean_compactness
-6.66	1	-33.7

mean_radius	mean_area
-111	1

mean_compactness	mean_concavity
-1.06	1

mean_concavity	mean_concavepoints
-0.445	1

mean_fractaldim	worst_fractaldim
1	-0.292

SE_radius	SE_perimeter
-6.88	1

SE_radius	SE_area
-151	1

SE_compactness	SE_concavity
-0.886	1

SE_compactness	SE_fractaldim
-0.0931	1

mean_radius	worst_radius
-1.3	1

mean_texture	worst_texture
-1.27	1

worst_radius	worst_perimeter
-6.63	1

worst_radius	worst_area
-139	1

worst_compactness	worst_fractaldim
1	-6.34



IDeACor <- cor(IDeAwpbc[,colnames(IDeAwpbc)!="status"])


  gplots::heatmap.2(abs(IDeACor),
                    trace = "none",
  #                  scale = "row",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "IDeA Correlation",
                    cexRow = 0.5,
                    cexCol = 0.5,
                     srtCol=45,
                     srtRow= -45,
                    key.title=NA,
                    key.xlab="Pearson Correlation",
                    xlab="Feature", ylab="Feature")



diag(IDeACor) <- 0;
getMaxcor <- apply(IDeACor,2,max)
topCorrelated <- getMaxcor[which.max(getMaxcor)]
whotoMax <- getMaxcor[getMaxcor == topCorrelated]
plot(IDeAwpbc[,names(whotoMax)],main="IDeA: Top Correlated variables")

plot(IDeAwpbc[,c("mean_radius","La_mean_perimeter")],main="IDeA: Top Raw Correlated variables")

1.3.3 PCA

featuresnames <- colnames(wpbc)[colnames(wpbc) != "status"]
pc <- prcomp(wpbc[,featuresnames],center = TRUE,tol=0.01)   #principal components
PCAwpbc <- as.data.frame(cbind(predict(pc,wpbc[,featuresnames]),status=wpbc$status))
#plot(PCAwpbc[,colnames(PCAwpbc)!="status"],col=wpbc$status,cex=0.65,cex.lab=0.5,cex.axis=0.75,cex.sub=0.5,cex.main=0.75)

#pander::pander(pc$rotation)


PCACor <- cor(PCAwpbc[,colnames(PCAwpbc)!="status"])


  gplots::heatmap.2(abs(PCACor),
                    trace = "none",
  #                  scale = "row",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "PCA Correlation",
                    cexRow = 0.5,
                    cexCol = 0.5,
                     srtCol=45,
                     srtRow= -45,
                    key.title=NA,
                    key.xlab="Pearson Correlation",
                    xlab="Feature", ylab="Feature")

1.3.4 EFA

featuresnames <- colnames(wpbc)[colnames(wpbc) != "status"]

uls <- fa(wpbc[,featuresnames],length(colnames(PCAwpbc)),rotate="varimax",warnings=FALSE)  # EFA analysis
EFAwpbc <- as.data.frame(cbind(predict(uls,wpbc[,featuresnames]),status=wpbc$status))
#plot(EFAwpbc[,colnames(EFAwpbc)!="status"],col=wpbc$status,cex=0.65,cex.lab=0.5,cex.axis=0.75,cex.sub=0.5,cex.main=0.75)
#pander::pander(uls$weights)


EFACor <- cor(EFAwpbc[,colnames(EFAwpbc)!="status"])


  gplots::heatmap.2(abs(EFACor),
                    trace = "none",
  #                  scale = "row",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "EFA Correlation",
                    cexRow = 0.5,
                    cexCol = 0.5,
                     srtCol=45,
                     srtRow= -45,
                    key.title=NA,
                    key.xlab="Pearson Correlation",
                    xlab="Feature", ylab="Feature")

1.4 Effect on CAR modeling

par(op)
par(xpd = TRUE)
wpbc$status <- factor(wpbc$status)
rawmodel <- rpart(status~.,wpbc,control=rpart.control(maxdepth=2))
plot(rawmodel,main="Raw",branch=0.5,uniform = TRUE,compress = TRUE,margin=0.1)
text(rawmodel, use.n = TRUE,cex=0.75)

pr <- predict(rawmodel,wpbc,type = "class")
pander::pander(table(wpbc$status,pr))

	0	1
0	95	8
1	19	15

pander::pander(c(accuracy=sum(wpbc$status==pr)/nrow(wpbc)))

accuracy
0.803


IDeAwpbc$status <- factor(IDeAwpbc$status)
IDeAmodel <- rpart(status~.,IDeAwpbc,control=rpart.control(maxdepth=2))
plot(IDeAmodel,main="IDeA",branch=0.5,uniform = TRUE,compress = TRUE,margin=0.1)
text(IDeAmodel, use.n = TRUE,cex=0.75)

pr <- predict(IDeAmodel,IDeAwpbc,type = "class")
pander::pander(table(IDeAwpbc$status,pr))

	0	1
0	103	0
1	24	10

pander::pander(c(accuracy=sum(IDeAwpbc$status==pr)/nrow(wpbc)))

accuracy
0.825



PCAwpbc$status <- factor(PCAwpbc$status)
PCAmodel <- rpart(status~.,PCAwpbc,control=rpart.control(maxdepth=2))
plot(PCAmodel,main="PCA",branch=0.5,uniform = TRUE,compress = TRUE,margin=0.1)
text(PCAmodel, use.n = TRUE,cex=0.75)

pr <- predict(PCAmodel,PCAwpbc,type = "class")
pander::pander(table(PCAwpbc$status,pr))

	0	1
0	98	5
1	25	9

pander::pander(c(accuracy=sum(PCAwpbc$status==pr)/nrow(wpbc)))

accuracy
0.781



EFAwpbc$status <- factor(EFAwpbc$status)
EFAmodel <- rpart(status~.,EFAwpbc,control=rpart.control(maxdepth=2))
plot(EFAmodel,main="EFA",branch=0.5,uniform = TRUE,compress = TRUE,margin=0.1)
text(EFAmodel, use.n = TRUE,cex=0.75)

pr <- predict(EFAmodel,EFAwpbc,type = "class")
pander::pander(table(EFAwpbc$status,pr))

	0	1
0	99	4
1	26	8

pander::pander(c(accuracy=sum(EFAwpbc$status==pr)/nrow(wpbc)))

accuracy
0.781


par(op)

1.5 Effect on logisitc modeling

wpbc$status <- 1*(wpbc$status == 1)
IDeAwpbc$status <- wpbc$status
PCAwpbc$status <- wpbc$status
EFAwpbc$status <- wpbc$status
rawmodel <- LASSO_MIN(status~.,wpbc,family="binomial")
pander::pander(rawmodel$coef)

(Intercept)	mean_area	mean_fractaldim	worst_perimeter	pnodes
-1.98	0.000569	-15.5	0.00826	0.0247


IDeAmodel <- LASSO_MIN(status~.,IDeAwpbc,family="binomial")
pander::pander(IDeAmodel$coef)

(Intercept)	mean_radius	La_mean_perimeter	La_mean_concavity	La_mean_fractaldim	SE_radius	La_SE_fractaldim	tsize	pnodes
-0.437	0.129	0.276	2.41	-38.5	0.095	-133	0.0169	0.0208



PCAmodel <- LASSO_MIN(status~.,PCAwpbc,family="binomial")
pander::pander(PCAmodel$coef)

(Intercept)	PC1	PC2
-1.2	-0.000887	0.00136



EFAmodel <- LASSO_MIN(status~.,EFAwpbc,family="binomial")
pander::pander(EFAmodel$coef)

(Intercept)	MR1
-1.15	0.433


par(op)

1.5.0.1 Standarize the names for the reporting

studyName <- "Wisconsin"
dataframe <- wpbc
outcome <- "status"
thro <- 0.80
TopVariables <- 10
cexheat = 0.25

1.6 Generaring the report

1.6.1 Libraries

Some libraries

library(psych)
library(whitening)
library("vioplot")

1.6.2 Data specs

pander::pander(c(rows=nrow(dataframe),col=ncol(dataframe)-1))

rows	col
137	32

pander::pander(table(dataframe[,outcome]))

0	1
103	34


varlist <- colnames(dataframe)
varlist <- varlist[varlist != outcome]

largeSet <- length(varlist) > 1500

1.6.3 Scaling the data

Scaling and removing near zero variance columns and highly co-linear(r>0.99999) columns


  ### Some global cleaning
  sdiszero <- apply(dataframe,2,sd) > 1.0e-16
  dataframe <- dataframe[,sdiszero]

  varlist <- colnames(dataframe)[colnames(dataframe) != outcome]
  tokeep <- c(as.character(correlated_Remove(dataframe,varlist,thr=0.99999)),outcome)
  dataframe <- dataframe[,tokeep]

  varlist <- colnames(dataframe)
  varlist <- varlist[varlist != outcome]


dataframe <- FRESAScale(dataframe,method="OrderLogit")$scaledData

1.7 The heatmap of the data


if (!largeSet)
{
  
  hm <- heatMaps(data=dataframe,
                 Outcome=outcome,
                 Scale=TRUE,
                 hCluster = "row",
                 xlab="Feature",
                 ylab="Sample",
                 srtCol=45,
                 srtRow=45,
                 cexCol=cexheat,
                 cexRow=cexheat
                 )
  par(op)
}

1.7.0.1 Correlation Matrix of the Data

The heat map of the data


if (!largeSet)
{

  par(cex=0.6,cex.main=0.85,cex.axis=0.7)
  #cormat <- Rfast::cora(as.matrix(dataframe[,varlist]),large=TRUE)
  cormat <- cor(dataframe[,varlist],method="pearson")
  cormat[is.na(cormat)] <- 0
  gplots::heatmap.2(abs(cormat),
                    trace = "none",
  #                  scale = "row",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "Original Correlation",
                    cexRow = cexheat,
                    cexCol = cexheat,
                     srtCol=45,
                     srtRow=45,
                    key.title=NA,
                    key.xlab="Pearson Correlation",
                    xlab="Feature", ylab="Feature")
  diag(cormat) <- 0
  print(max(abs(cormat)))
}

[1] 0.9961165

1.8 The decorrelation


DEdataframe <- IDeA(dataframe,verbose=TRUE,thr=thro)
#> 
#>  Included: 32 , Uni p: 0.02425652 , Uncorrelated Base: 8 , Outcome-Driven Size: 0 , Base Size: 8 
#> 
#> 
 1 <R=0.996,r=0.973,N=    6>, Top: 2( 2 )[ 1 : 2 Fa= 2 : 0.973 ]( 2 , 4 , 0 ),<|>Tot Used: 6 , Added: 4 , Zero Std: 0 , Max Cor: 0.968
#> 
 2 <R=0.968,r=0.959,N=    6>, Top: 1( 1 )[ 1 : 1 Fa= 3 : 0.959 ]( 1 , 1 , 2 ),<|>Tot Used: 8 , Added: 1 , Zero Std: 0 , Max Cor: 0.928
#> 
 3 <R=0.928,r=0.914,N=    4>, Top: 2( 1 )[ 1 : 2 Fa= 3 : 0.914 ]( 2 , 2 , 3 ),<|>Tot Used: 9 , Added: 2 , Zero Std: 0 , Max Cor: 0.893
#> 
 4 <R=0.893,r=0.846,N=   10>, Top: 5( 1 )[ 1 : 5 Fa= 8 : 0.846 ]( 5 , 5 , 3 ),<|>Tot Used: 18 , Added: 5 , Zero Std: 0 , Max Cor: 0.837
#> 
 5 <R=0.837,r=0.819,N=   10>, Top: 3( 1 )[ 1 : 3 Fa= 8 : 0.819 ]( 3 , 3 , 8 ),<|>Tot Used: 20 , Added: 3 , Zero Std: 0 , Max Cor: 0.783
#> 
 6 <R=0.783,r=0.800,N=   10>
#> 
 [ 6 ], 0.7809466 Decor Dimension: 20 Nused: 20 . Cor to Base: 11 , ABase: 2 , Outcome Base: 0 
#> 
varlistc <- colnames(DEdataframe)[colnames(DEdataframe) != outcome]

pander::pander(sum(apply(dataframe[,varlist],2,var)))

32.9

pander::pander(sum(apply(DEdataframe[,varlistc],2,var)))

21.8

pander::pander(entropy(discretize(unlist(dataframe[,varlist]), 256)))

4.95

pander::pander(entropy(discretize(unlist(DEdataframe[,varlistc]), 256)))

4.66

1.8.1 The decorrelation matrix


if (!largeSet)
{

  par(cex=0.6,cex.main=0.85,cex.axis=0.7)
  
  UPSTM <- attr(DEdataframe,"UPSTM")
  
  gplots::heatmap.2(1.0*(abs(UPSTM)>0),
                    trace = "none",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "Decorrelation matrix",
                    cexRow = cexheat,
                    cexCol = cexheat,
                   srtCol=45,
                   srtRow=45,
                    key.title=NA,
                    key.xlab="|Beta|>0",
                    xlab="Output Feature", ylab="Input Feature")
  
  par(op)
}

1.9 The heatmap of the decorrelated data

if (!largeSet)
{

  hm <- heatMaps(data=DEdataframe,
                 Outcome=outcome,
                 Scale=TRUE,
                 hCluster = "row",
                 cexRow = cexheat,
                 cexCol = cexheat,
                 srtCol=45,
                 srtRow=45,
                 xlab="Feature",
                 ylab="Sample")
  par(op)
}

1.10 The correlation matrix after decorrelation

if (!largeSet)
{

  cormat <- cor(DEdataframe[,varlistc],method="pearson")
  cormat[is.na(cormat)] <- 0
  
  gplots::heatmap.2(abs(cormat),
                    trace = "none",
                    mar = c(5,5),
                    col=rev(heat.colors(5)),
                    main = "Correlation after IDeA",
                    cexRow = cexheat,
                    cexCol = cexheat,
                     srtCol=45,
                     srtRow=45,
                    key.title=NA,
                    key.xlab="Pearson Correlation",
                    xlab="Feature", ylab="Feature")
  
  par(op)
  diag(cormat) <- 0
  print(max(abs(cormat)))
}

[1] 0.7827266

1.11 U-MAP Visualization of features

1.11.1 The UMAP based on LASSO on Raw Data

classes <- unique(dataframe[,outcome])
raincolors <- rainbow(length(classes))
names(raincolors) <- classes
datasetframe.umap = umap(scale(dataframe[,varlist]),n_components=2)
plot(datasetframe.umap$layout,xlab="U1",ylab="U2",main="UMAP: Original",t='n')
text(datasetframe.umap$layout,labels=dataframe[,outcome],col=raincolors[dataframe[,outcome]+1])

1.11.2 The decorralted UMAP


datasetframe.umap = umap(scale(DEdataframe[,varlistc]),n_components=2)
plot(datasetframe.umap$layout,xlab="U1",ylab="U2",main="UMAP: After IDeA",t='n')
text(datasetframe.umap$layout,labels=DEdataframe[,outcome],col=raincolors[DEdataframe[,outcome]+1])

1.12 Univariate Analysis

1.12.1 Univariate



univarRAW <- uniRankVar(varlist,
               paste(outcome,"~1"),
               outcome,
               dataframe,
               rankingTest="AUC")



univarDe <- uniRankVar(varlistc,
               paste(outcome,"~1"),
               outcome,
               DEdataframe,
               rankingTest="AUC",
               )

1.12.2 Final Table


univariate_columns <- c("caseMean","caseStd","controlMean","controlStd","controlKSP","ROCAUC")

##topfive
topvar <- c(1:length(varlist)) <= TopVariables
pander::pander(univarRAW$orderframe[topvar,univariate_columns])

	caseMean	caseStd	controlMean	controlStd	controlKSP	ROCAUC
mean_perimeter	0.489	0.908	-0.13157	0.779	4.26e-01	0.697
mean_area	0.556	0.974	-0.09535	0.782	9.68e-02	0.696
mean_radius	0.486	0.904	-0.13511	0.801	3.28e-01	0.693
worst_radius	0.570	0.949	-0.06074	0.822	2.93e-01	0.691
worst_perimeter	0.591	0.977	-0.05457	0.808	5.49e-01	0.691
worst_area	0.656	1.077	0.00399	0.852	2.72e-01	0.683
tsize	0.591	0.988	0.08954	1.116	3.18e-03	0.680
mean_fractaldim	-0.252	0.687	0.22329	0.927	6.06e-01	0.657
pnodes	1.014	1.373	0.36684	1.140	2.35e-08	0.655
SE_area	0.594	1.033	0.09745	0.869	3.92e-02	0.653



topLAvar <- univarDe$orderframe$Name[str_detect(univarDe$orderframe$Name,"La_")]
topLAvar <- unique(c(univarDe$orderframe$Name[topvar],topLAvar[1:as.integer(TopVariables/2)]))
finalTable <- univarDe$orderframe[topLAvar,univariate_columns]

theLaVar <- rownames(finalTable)[str_detect(rownames(finalTable),"La_")]

pander::pander(univarDe$orderframe[topLAvar,univariate_columns])

	caseMean	caseStd	controlMean	controlStd	controlKSP	ROCAUC
mean_radius	0.48605	0.9044	-0.13511	0.8011	3.28e-01	0.693
tsize	0.59083	0.9883	0.08954	1.1155	3.18e-03	0.680
pnodes	1.01412	1.3734	0.36684	1.1399	2.35e-08	0.655
SE_area	0.59388	1.0326	0.09745	0.8689	3.92e-02	0.653
La_mean_fractaldim	-0.13729	0.2637	0.06482	0.4889	7.68e-01	0.644
La_mean_concavity	0.14360	0.5879	-0.13548	0.4464	8.26e-01	0.635
La_mean_concavepoints	0.12493	0.4004	-0.03533	0.4642	7.48e-01	0.614
La_worst_compactness	0.20999	0.7221	-0.05690	0.6046	7.11e-01	0.607
La_mean_perimeter	0.00251	0.0337	-0.00841	0.0336	6.07e-02	0.603
La_SE_fractaldim	-0.16095	0.5129	0.06813	0.5573	3.05e-01	0.603


dc <- getLatentCoefficients(DEdataframe)
fscores <- attr(DEdataframe,"fscore")

theSigDc <- dc[theLaVar]
names(theSigDc) <- NULL
theSigDc <- unique(names(unlist(theSigDc)))


theFormulas <- dc[rownames(finalTable)]
deFromula <- character(length(theFormulas))
names(deFromula) <- rownames(finalTable)

pander::pander(c(mean=mean(sapply(dc,length)),total=length(dc),fraction=length(dc)/(ncol(dataframe)-1)))

mean	total	fraction
2.07	14	0.438



allSigvars <- names(dc)



dx <- names(deFromula)[1]
for (dx in names(deFromula))
{
  coef <- theFormulas[[dx]]
  cname <- names(theFormulas[[dx]])
  names(cname) <- cname
  for (cf in names(coef))
  {
    if (cf != dx)
    {
      if (coef[cf]>0)
      {
        deFromula[dx] <- paste(deFromula[dx],
                               sprintf("+ %5.3f*%s",coef[cf],cname[cf]))
      }
      else
      {
        deFromula[dx] <- paste(deFromula[dx],
                               sprintf("%5.3f*%s",coef[cf],cname[cf]))
      }
    }
  }
}

finalTable <- rbind(finalTable,univarRAW$orderframe[theSigDc[!(theSigDc %in% rownames(finalTable))],univariate_columns])


orgnamez <- rownames(finalTable)
orgnamez <- str_remove_all(orgnamez,"La_")
finalTable$RAWAUC <- univarRAW$orderframe[orgnamez,"ROCAUC"]
finalTable$DecorFormula <- deFromula[rownames(finalTable)]
finalTable$fscores <- fscores[rownames(finalTable)]

Final_Columns <- c("DecorFormula","caseMean","caseStd","controlMean","controlStd","controlKSP","ROCAUC","RAWAUC","fscores")

finalTable <- finalTable[order(-finalTable$ROCAUC),]
pander::pander(finalTable[,Final_Columns])

	DecorFormula	caseMean	caseStd	controlMean	controlStd	controlKSP	ROCAUC	RAWAUC	fscores
mean_perimeter	NA	0.48901	0.9082	-0.13157	0.7788	4.26e-01	0.697	0.697	NA
mean_radius		0.48605	0.9044	-0.13511	0.8011	3.28e-01	0.693	0.693	3
tsize		0.59083	0.9883	0.08954	1.1155	3.18e-03	0.680	0.680	NA
mean_fractaldim	NA	-0.25171	0.6868	0.22329	0.9273	6.06e-01	0.657	0.657	NA
pnodes		1.01412	1.3734	0.36684	1.1399	2.35e-08	0.655	0.655	NA
SE_area		0.59388	1.0326	0.09745	0.8689	3.92e-02	0.653	0.653	2
La_mean_fractaldim	+ 1.000mean_fractaldim -0.759worst_fractaldim	-0.13729	0.2637	0.06482	0.4889	7.68e-01	0.644	0.657	-1
La_mean_concavity	-0.706mean_compactness + 1.000mean_concavity	0.14360	0.5879	-0.13548	0.4464	8.26e-01	0.635	0.595	0
mean_concavepoints	NA	0.38302	1.0586	-0.06750	0.9633	2.98e-01	0.619	0.619	NA
La_mean_concavepoints	-0.992mean_concavity + 1.000mean_concavepoints	0.12493	0.4004	-0.03533	0.4642	7.48e-01	0.614	0.619	-1
La_worst_compactness	+ 1.000worst_compactness -0.968worst_fractaldim	0.20999	0.7221	-0.05690	0.6046	7.11e-01	0.607	0.516	-1
La_mean_perimeter	-0.980mean_radius + 1.000mean_perimeter -0.063*mean_compactness	0.00251	0.0337	-0.00841	0.0336	6.07e-02	0.603	0.697	-2
La_SE_fractaldim	-0.873SE_compactness + 1.000SE_fractaldim	-0.16095	0.5129	0.06813	0.5573	3.05e-01	0.603	0.549	-1
worst_fractaldim	NA	-0.15068	0.8304	0.20868	1.0616	3.22e-01	0.598	0.598	2
mean_concavity	NA	0.26012	0.9174	-0.03242	0.8892	6.78e-01	0.595	0.595	NA
SE_fractaldim	NA	-0.01848	0.8634	0.23441	1.1722	4.15e-01	0.549	0.549	NA
SE_compactness	NA	0.16325	1.1390	0.19053	1.0887	2.15e-01	0.525	0.525	2
worst_compactness	NA	0.06410	1.0995	0.14515	1.2064	3.09e-01	0.516	0.516	NA
mean_compactness	NA	0.16493	0.9543	0.14588	1.1026	5.38e-01	0.509	0.509	2

Decorrelation-Based Feature Discovery: Wisconsin

Jose Tamez

2023-09-08